Suppose you have a dispute with a neighbor. You can't reach an agreement on your own- each of you thinks he is right and refuses to budge. Do you start beating the neighbor with a rake? Well, hopefully not.
The common solution is to seek arbitration.
Now, there is no "only way" this is done, but there are some fairly consistent trends, and this is what I'll describe to you now.
Either you choose a mutually-satisfactory arbitrator, or you allow your respective representatives (you do have one, right?) to get together to hire one for the case. There are many who offer arbitration services full-time, and even more people who have gained a reputation for fairness and wisdom who are sought out even though it is not their "real job". Any fees are paid up front.
If a person simply refuses to enter arbitration, or will not agree to any arbitrator that is suggested, they have already "lost". I'll discuss the "loser" later.
The arbitrator will hear both sides, and do any investigation he deems necessary (either personally or through investigators employed for that specific purpose).
Rights, and respect for liberty are the top considerations. Contracts and other agreements fall only slightly below those- and it will be considered whether those agreements violate natural rights in any way. That isn't always a deal-breaker, though.
The decision is then rendered, and the parties will have already agreed to abide by the decision as a part of signing on for the arbitration.
Now, there is no "enforcement" of a decision. So how do you make the "loser" follow through? You don't. However, failing to abide by the decision is a huge strike against an individual's reputation, and reputation is very important in Libertopia.
The "winner" who is left holding the bag is perfectly free to spread the word about the failure of the "loser" to live up to his obligation. Most people will choose to avoid entering into deals with people who are known to not be trustworthy. Or, they will charge more due to perceived risk.
And if you think it is getting hard to hide from your past in the early days of the internet, wait til you get here! Sure, it can be done, but it is complicated and inconvenient enough that it will still usually remove the uncooperative fellow from your life so that at least he won't be bothering you anymore. And even if he runs far away, his past will probably still come back to haunt him since a man without an established reputation will find it hard to do business with anyone (without, once again, paying a premium).
Is this perfect? No. Fallible human beings are involved. Is it the best system yet devised? I certainly think so, and so do a lot of others apparently, since it is all completely voluntary and very few choose to opt out.